Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tilings of the Sphere by Congruent Pentagons IV: Edge Combination $a^4b$

Hoi Ping Luk, Min Yan

Abstract

We classify edge-to-edge tilings of the sphere by congruent almost equilateral pentagons, in which four edges have the same length. Together with our earlier classifications of edge-to-edge tilings of the sphere by congruent equilateral pentagons of other types, and our classification of edge-to-edge tilings of the sphere by congruent quadrilaterals or triangles, we complete the classification of edge-to-edge tilings of the sphere by congruent polygons.

Tilings of the Sphere by Congruent Pentagons IV: Edge Combination $a^4b$

Abstract

We classify edge-to-edge tilings of the sphere by congruent almost equilateral pentagons, in which four edges have the same length. Together with our earlier classifications of edge-to-edge tilings of the sphere by congruent equilateral pentagons of other types, and our classification of edge-to-edge tilings of the sphere by congruent quadrilaterals or triangles, we complete the classification of edge-to-edge tilings of the sphere by congruent polygons.
Paper Structure (15 sections, 50 theorems, 256 equations, 74 figures)

This paper contains 15 sections, 50 theorems, 256 equations, 74 figures.

Table of Contents

  1. Introduction
  2. Construction
  3. Technique
  4. Notations and Conventions
  5. Special Tile and Companion Pair
  6. Counting
  7. Geometry
  8. Existence of Pentagon
  9. Symmetric Pentagon
  10. $\alpha,\beta,\gamma$ Are Distinct
  11. $\alpha\delta\epsilon$ and One Degree $3$ $\hat{b}$-Vertex
  12. $\beta\delta\epsilon$ and One Degree $3$ $\hat{b}$-Vertex
  13. Two Matched Degree $3$ $b$-Vertices
  14. Single Matched Degree $3$ $b$-Vertex
  15. $\alpha,\beta,\gamma$ Are Not Distinct

Key Result

Theorem 1

Edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ ($a,b$ distinct) are the following:

Figures (74)

  • Figure 1: Almost equilateral pentagon.
  • Figure 2: Pentagonal subdivision tilings for $a^4b$.
  • Figure 3: Earth map tilings for $a^4b$.
  • Figure 4: Flips modifications $F_1E_{\pentagon}1,F_2E_{\pentagon}1$ of $E_{\pentagon}1$.
  • Figure 5: Flip modification of $E_{\pentagon}2$.
  • ...and 69 more figures

Theorems & Definitions (92)

  • Theorem
  • Lemma 1: Parity Lemma
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5: Counting Lemma
  • proof
  • Lemma 6: Balance Lemma
  • proof
  • ...and 82 more